Showing $supp{f} =\overline{\{x \in X: f(x) \neq 0\}}$

Let $ f: \mathbb R^{d} \to \mathbb R$

We define the support of $ f$ as $ supp f=(\bigcup_{A \subseteq \mathbb R^{d} open,f\vert_{A}=0, \lambda^{d}-a.e.}A)^{c}$

How can I show for $ f \in C(\mathbb R^{d})$ that $ supp{f} =\overline{\{x \in X: f(x) \neq 0\}}$

How do I go about showing $ (\bigcup_{A \subseteq \mathbb R^{d} open,f\vert_{A}=0, \lambda^{d}-a.e.}A)^{c}\subseteq \overline{\{x \in X: f(x) \neq 0\}}$

and conversely,

$ \overline{\{x \in X: f(x) \neq 0\}}\subseteq (\bigcup_{A \subseteq \mathbb R^{d} open,f\vert_{A}=0, \lambda^{d}-a.e.}A)^{c}$